$$A$$ vector is defined as $$f = y\widehat i + x\widehat j + z\widehat k\,\,$$. Where $$\widehat i,\widehat j,$$ and $$\widehat k$$ are unit vectors in cartesian $$(x, y, z)$$ coordinate system. The surface integral - -over the closed surface $$S$$ of a cube with vertices having the following coordinates: $$(0,0,0), (1, 0, 0), (0, 1, 0), (0,0,1), (1, 0, 1), (1,1,1), (0, 1, 1), (1, 1, 0)$$ is _______.

Given that $$x$$ is a random variable in the range $$\left[ {0,\infty } \right]$$ with a probability density function $${{{e^{ - {x \over 2}}}} \over K},$$ the value of the constant $$K$$ is _______

The figure shown the schematic of a production process with machines $$A, B$$ and $$C.$$ An input job needs to be pre-processed either by $$A$$ or by $$B$$ before it is fed to $$C,$$ from which the final finished product comes out. The probabilities of failure of the machines are given as:
$${P_{\rm A}} = 0.15,\,\,{P_{\rm B}} = 0.05\,\,\& \,\,{P_C} = 0.1$$

Assuming independence of failures of the machines, the probability that a given job is successfully processed (up to the third decimal place) is _____.

The figure shows the plot of $$y$$ as a function of $$x$$

The function shown in the solution of the differential equation (assuming all initial conditions to be zero) is